Table of Contents

## Symmetric matrix

Definition of a symmetric matrix: A square matrix A = [a_{ij}] is called a symmetric matrix if a_{ij} = a_{ij}, for all i,j values

**Example of the symmetric matrix:**

is an example of a symmetric matrix

Note: Matrix A is symmetric if A’ = A (where A’ is the transpose of the matrix)

### Transpose of matrices

To understand if a matrix is a symmetric matrix, it is very important to know about the transpose of a matrix and how to find the transpose of a matrix.

If the rows and columns of an m×n matrix are interchanged to get an n × m matrix, the new matrix obtained is called the transpose of the given matrix.

## Skew-Symmetric Matrix

**Definition of Skew-symmetric matrix:** A square matrix A = [aij] is a skew-symmetric matrix if aij = -aji, for all values of i,j.

If we put i=j, then,

a_{ii} = -a_{ii}

⇒ 2a_{ii} = 0

⇒ a_{ii} = 0

Thus, **in a skew-symmetric matrix, all diagonal elements are zero.**

**Example of a Skew-symmetric matrix:**

is an example of skew-symmetric matrices.

**Note:** A square matrix A is skew-symmetric if A’ = -A.

**Properties of Symmetric and skew-symmetric matrices**

- Every square Matrix can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix.
- If A is a symmetric Matrix, then A
^{n}, n belonging to the Natural number, will be symmetric. - A is skew-symmetric, then A
^{n}, n N will be symmetric if n is even, and A^{n}will be skew-symmetric if n is odd. - Null Matrix is always symmetric and skew-symmetric.
- If A and B are both symmetric, then AB + BA will be symmetric, and AB – BA will be skew-symmetric.

### What is the difference between a symmetric matrix and a skew-symmetric matrix?

A symmetric matrix and a skew-symmetric matrix are the square matrices. But the major difference between them is:

- The symmetric matrix equals its transpose.

⟹If A is a symmetric matrix, then A = A^{T} - Whereas a skew-symmetric matrix is a matrix whose transpose equals its negative.

⟹If A is a skew-symmetric matrix, then A^{T}= – A.

For better understanding of matrices, also read:

Matrices | Eigenvalues of a symmetric matrix | Inverse of matrices |

Determinants | Transpose of matrices | Types of matrices |

## FAQs on Symmetric and Skew-symmetric Matrix

### Can a matrix be both symmetric and skew-symmetric?

No, a matrix cannot be symmetric and skew-symmetric unless it is the null matrix (a matrix with zero elements). In a non-null matrix, the presence of non-zero diagonal elements in a symmetric matrix contradicts the property of having zero diagonal elements in a skew-symmetric matrix.

### Are symmetric and skew-symmetric matrices always square matrices?

Both symmetric and skew-symmetric matrices are defined for square matrices only. In other words, their number of rows equals the number of columns.

### Are the eigenvalues of a symmetric matrix real?

Yes, all eigenvalues of a symmetric matrix are real. This property is known as the spectral theorem for symmetric matrices.

### What is the relationship between the eigenvectors of a symmetric matrix?

The eigenvectors corresponding to distinct eigenvalues of a symmetric matrix are orthogonal (perpendicular) to each other.

### Can a matrix be both symmetric and diagonal?

A diagonal matrix where all non-diagonal elements are zero is symmetric and diagonal.

### Can a matrix be both skew-symmetric and diagonal?

No, a matrix cannot be skew-symmetric and diagonal unless it is the null matrix. In a non-null matrix, the presence of non-zero diagonal elements contradicts the property of having zero diagonal elements in a skew-symmetric matrix.

### How can we determine if a matrix is symmetric or skew-symmetric?

To determine if a matrix is symmetric, we compare it to its transpose. If the matrix is equal to its transpose (A = A'), it is symmetric. To determine if a matrix is skew-symmetric, we compare it to the negation of its transpose. If the matrix equals the negation of its transpose (A = -A'), it is skew-symmetric.